On 10 June 2010 16:52, David Goldsmith <d.l.goldsmith@gmail.com> wrote:
> I recently had cause to ponder moving averages, and given my general
> interest in noise theory, it got me wondering: relative to the PS of the
> signal, what's the PS of a n-width moving average. After unsuccessfully
> (though far from exhaustively) looking for some results in the literature, I
> just started thinking about it myself, and came to realize, both based on
> what the graphs are saying about the situation and then in light of that, in
> retrospect, conceptually as well, since the moving average is a smoothing of
> the signal, it's some kind of low-pass filter (removing power at higher
> frequencies), which begs the question: what kind of low-pass filter? In
> particular, is it a truncation filter, completely removing any power from
> windows smaller than n (the intuitive, though far from obvious, conclusion),
> or is it an attenuation filter, applying some monotonically decreasing
> envelope to the PS for frequencies corresponding to windows smaller than n?
> (Or does it somehow influence even the power of frequencies corresponding to
> windows larger than n?) Reference/proof? Thanks for the education.
An n-width moving average is (I'm assuming equally-spaced data points)
convolution by a boxcar of width n. So its effect on the power
spectrum is multiplication by a sinc function whose first zero is at a
period of n samples and whose amplitude at zero frequency is 1. (If
you have a finite-length data set and are doing circular convolution,
for "sinc" read the Dirichlet kernel.)
Anne
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